It is difficult to reduce the memory requirements of solving the Tower of
Hanoi problem without significantly
increasing the running time of the solver on a single machine.
However, we have come up with a distributed solution for Tower of Hanoi. The
idea is to arrange n-computers into a ring, where:

x computers store the previous move data,

y computers store the current move data, and

z computers store the next move data

with
x + y + z = n.

This distributed solution is theoretically scalable to any
number of computers - allowing one to postpone the point at which memory
problems overwhelm the solver.

We estimate that, given a dozen computers with at least 1 GB
of RAM each, one could figure out up to ToH(4,22) in a week or so of
processing time. As a proof of concept, we implemented a prototype distributed
solver, which can run on 3 machines at once. (For decent speed it requires at
least 100Mbit Ethernet, and preferably gigabit Ethernet.)

Algorithm for Three Machines

Given three machines, we assign each one a distinct role: an
Sn-1 client, an Sn client and an
Sn+1 client.

The operation of the cluster is as follows:

Startup The
Sn client will be initially seeded with the starting state. Both
the Sn-1 and the Sn+1 clients will have their stores empty.

Step 1 The Sn client generates all the adjacent states to the states in its
store and sends those new states to the Sn-1 client. The
Sn-1 client
compares all the states it receives from the Sn client against its store
and only sends the ones that are not present in its store to the Sn+1
client. The Sn+1 client adds the states it
receives from the Sn-1 client to its store if the
states are not already present.

Step 2 After
the Sn client has send all adjacent states to the Sn-1
client, the Sn-1 client has sent its last state to
the Sn+1 client and the Sn+1
client has processed the final state from Sn-1, the
clients will rename themselves using the following mapping: Sn+1→Sn, Sn→Sn-1, Sn-1→Sn+1.
Now go to step 1 if a halting state has not yet been found.

Figure 1: The inter-machine Tower of Hanoi state flow across three
complete iterations of the three machine algorithm.